# Exponentials and Logarithms

A collection of resources to support the teaching of indices, exponentials and logaritms

### Algebra

These resources cover a wide range of algebraic topics, many of which are suitable for students studying mathematics at Higher Level GCSE, or A Level, as well as those students for whom mathematics is an integral part of their course. Some of the topics covered include completing the square, factorising quadratics, partial fractions, integration, simultaneous linear equations, logarithms and polynomial division.

Comprehensive notes, with clear descriptions, for each resource are provided, together with relevant diagrams and examples. Students wishing to review, and consolidate, their knowledge and understanding of algebraic principles will find them useful, as each topic includes a selection of questions to be completed, for which answers are provided.

### Using Indices N12

This resource introduces fractional and negative indices to enable students to evaluate numerical expressions using negative and fractional

indices and use the rules of indices with integer and fractional powers of variables.

### Indices

This resource contains five activities designed to enable students to explore indices both numerically and algebraically. The resource features a number of activities dealing with negative indices and fractional indices.

Starter activities: contains two starter activities. In the first activity students have to represent an integer value in as many ways as they can, each way to include an index value. The second activity, ‘Why does?’ students have to explain mathematical statements involving indices.

Matching pairs game: Students are required to match a number in index form with its integer value.

Ordering indices: Students compare the values written in index form and place the cards in numerical order.

Odd one out: Given three cards, students find the odd one out and make up their own card to match the odd one.

True or false: Aimed at addressing common misconceptions, in this activity students discuss whether the statements given are true or false.

### Further Indices

This resource contains four activities designed to explore the rules of indices as well as differentiating and integrating functions containing indices.

The rules of indices with algebraic expressions: This activity is a matching exercise with algebraic statements involving negative indices and fractional indices.

Dominoes: is a loop card activity involving negative indices and fractional indices.

Differentiation and integration involving indices: A large, multi-tiered activity matching equivalent functions, differentials and integrals.

Marking: In this activity, students are given questions and solutions to a number of differentiation and integration questions. Students are required to mark the work. The solutions contain many common errors that are made. Students should mark the work for accuracy, correct solutions where necessary and give advice to help the candidate therefore explain what the error was and how to correct it.

### The Exponential Function

This interactive resource is designed to enable students to explore the nature of the exponential function and to explore the derivative of the exponential function.

An introduction page sets out some basic information about exponential functions leading to a definition of the exponential function. There follows an interactive graphical page showing graphically the definition of the exponential function. The next interactive graph enables students to find an approximation for the value of e. There follows a summary of the main points.

### Logarithms

This resource contains six problems which require deep understanding of the working of logarithms. Students are required to develop questions which give a specific answer. It is likely that each student will provide a different solution. This presents an opportunity for rich discussion as students question and justify the solutions presented as do the true/false questions.

### Logarithms

RISP activity Building Log Equations requires students to form equations given a set of cards and to determine, with examples, whether the equation is always, sometimes or never true and to attempt to say why. Students must include at least one log card in their equation. Students need to be familiar with logs in different number bases.

### Simplifying Logarithmic Expressions A13

This resource enables students to develop their understanding of the laws of logarithms, practise using the laws of logarithms to simplify numerical expressions involving logarithms and apply their knowledge of the laws of logarithms to expressions involving variables. Students should have some knowledge of the laws of logarithms as applied to numerical expressions

### Simplifying Expressions Using the Rules of Logarithms

This resource contains four advanced level activities designed to practise the basics and extend understanding of logarithms. There is also a set of open ended questions designed to assess understanding and inform further teaching. There is no need for any additional teaching before each activity – the student can work out the next stage from their existing knowledge.

Logarithms: true or false: pairs of students are presented with twelve statements and have to decide whether each statement is true or false and provide justification.

Logarithms: ordering: students have to place cards in order of value using logarithms that do not have simple rational answers and therefore must be estimated in order to compare values.

Logarithms: odd-one-out: students have to decide which of the three statements is the odd-one-out and then write an expression to match the odd-one-out. The activity contains numerical and algebraic values.

Logarithms: matching up: students have to focus on the laws of logarithms in order to match cards with equivalent values. Not all cards pair up, some have more than one partner and some have no partner at all.

Logarithms: open questions: a series of open-ended, probing questions, designed to assess students' knowledge, identify misconceptions and inform future teaching.

### Episode 513: Preparation for the Exponential Decay Topic

From the Institute of Physics, these activities are not concerned with the products of radioactive decay but look at the random, spontaneous nature of radioactive decay and its consequences.

Upon completion of the topic, students should be able to:

1. define the term half-life

2. make calculations involving numbers of half-lives

3. relate half-life to decay probability

4. measure the half-life of a fast-decaying nuclide

5. use exponential and logarithmic equations for radioactive decay.

### Episode 514: Patterns of Decay

In this learning episode, from the Institute of Physics, students use discussion, demonstration and worked examples to look at the exponential decay and radioactive half-life of a range of nuclides. These include sources of alpha, beta and gamma radiation.

The activities in this learning episode include:

• discussing and demonstrating half-life

• measuring the half-life of protactinium ^{234}Pa

• worked examples on whole numbers of half lives

• student examples and calculations

### Episode 515: The Radioactive Decay Formula

Produced by the Institute of Physics, this learning episode will help students look mathematically at the meaning and relevance of the decay constant. A simple model is used to illustrate radioactive decay.

### Episode 516: Exponential and Logarithmic Equations

In this learning episode, from the Institute of Physics, mathematical ideas on radioactive decay are developed using logarithmic and exponential equations.